Tutorial 4: Worksheet
Trigonometry. Recall that
y(t) = c1 cos(ωt) + c2 sin(ωt),
can be written in the form
y(t) = A cos(ωt − γ),
with A =
q
c21 + c22 , and γ = arctan
c2
c1
.
1. Free undamped motion. Consider a mass and spring system with a mass m = 2
and spring constant k = 3. Set up and find the general solution of the system.
2. Damped motion. Consider a mass and spring system with a mass m = 2, spring
constant k = 3, and damping constant c = 1. Set up and find the general solution of
the system.
Beats. The general solution of mx00 + kx = F0 cos(ωt), with ω 6= ω0 =
x(t) = c1 cos(ω0 t) + c2 sin(ω0 t) +
3. Find the general solution to
time t?
x00
2
is
F0
cos(ωt).
− ω2)
+ 8x = 10 cos(πt). Does x(t) remain bounded for all
x(t) = c1 cos(ωt) + c2 sin(ωt) +
x00
2
k
m
m(ω02
Resonance. The general solution of mx00 + kx = F0 cos(ωt), with ω = ω0 =
4. Find the general solution to
time t?
q
q
k
m
is
F0
t sin(ωt).
2mω
+ 8x = 10 cos(4t). Does x(t) remain bounded for all
5. Forced oscillations with damping. Find the steady state periodic solution to the
mass spring system described by x00 + cx0 + x = 10 cos(ωt), where ω and c are both
positive constants. Your answer should look like x(t) = A(ω) cos(ωt) + B(ω) sin(ωt).
6. Maximizing practical resonance. For what angular frequency ω is the amplitude
D(ω) of the steady state periodic solution you obtained in Question (5) maximized?
Under what conditions are your critical points actually maxima?
First hint: use the Trigonometry fact at the top of this sheet.
Second hint: Note that
1
g(ω)
has a maximum value when g(ω) is minimized and that
p
f (ω) has a minimum when f (w) has a minimum.
1